Algebra Universalis最新文献

Câmpeanu and Ho (2004) determined the maximum finite state complexity of finite languages, building on work of Champarnaud and Pin (1989). They stated that it is very difficult to determine the number of maximum-complexity languages. Here we give a formula for this number. We also generalize their work from languages to functions on finite sets.

In this paper, we shed new light on the spectrum of the relation algebra we call (A_{n}), which is obtained by splitting the non-flexible diversity atom of (6_{7}) into n symmetric atoms. Precisely, show that the minimum value in (text {Spec}(A_{n})) is at most (2n^{6 + o(1)}), which is the first polynomial bound and improves upon the previous bound due to Dodd and Hirsch (J Relat Methods Comput Sci 2:18–26, 2013). We also improve the lower bound to (2n^{2} + 4n + 1), which is roughly double the trivial bound of (n^{2} + 2n + 3). In the process, we obtain stronger results regarding (text {Spec}(A_{2}) =text {Spec}(32_{65})). Namely, we show that 1024 is in the spectrum, and no number smaller than 26 is in the spectrum. Our improved lower bounds were obtained by employing a SAT solver, which suggests that such tools may be more generally useful in obtaining representation results.

A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have the natural structure of a Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.

Let ({mathcal {A}}) be a class of algebras with (I, A in {mathcal {A}}). We interpret the lattice-theoretic “strictly meet irreducible/cover” situation (B < C) in lattices of the form (S_{{mathcal {A}}}(I,A)) of all subalgebras of A containing I, where we call such (B < C) a minimum proper extension (mpe), and show that this means B is maximal in (S_{{mathcal {A}}}(I,A)) for not containing some (r in A) and C is generated by B and r. For the class ({mathcal {G}}) of groups, we determine the mpe’s in (S_{{mathcal {G}}}({0},{mathbb {Q}})) using invariants of Beaumont and Zuckerman and show that these (plus utilization of a Hamel basis) determine the mpe’s in (S_{{mathcal {G}}}({0},{mathbb {R}})). Finally, we show that the latter yield some (not all) of the minimum proper essential extensions in (mathbf {W}^{*}), the category of Archimedean (ell )-groups with strong order unit and unit-preserving (ell )-group homomorphisms.

For a dynamical system (X, f), the notion of topological transitivity has been studied by some researchers. There are several definitions of this property, and it is part of the folklore of dynamical systems that under some hypotheses, they are equivalent. In this paper, our aim is to introduce and study some properties of topological transitivity in pointfree topology, for which we first need to define in a way what makes them conservative extensions of topological transitivity defined by G.D. Birkhoff. We describe the way the different properties are related to each other in pointfree topology.

Positive MV-algebras are the subreducts of MV-algebras with respect to the signature ({oplus , odot , vee , wedge , 0, 1}). We provide a finite quasi-equational axiomatization for the class of such algebras.

Based on the previously obtained concrete characterization of the endomorphism semigroups of quasi-acyclic reflexive graphs we prove the relatively elementary definability of the class of such graphs in the class of all semigroups. It will permit us to investigate for such graphs the abstract representation problem for the endomorphism semigroups of graphs and the problem of elementary definability of graphs by their endomorphism semigroups.

It was proved by the authors that the quasivariety of quasi-Stone algebras (mathbf {Q}_{mathbf {1,2}}) is finite-to-finite universal relative to the quasivariety (mathbf {Q}_{mathbf {2,1}}) contained in (mathbf {Q}_{mathbf {1,2}}). In this paper, we prove that (mathbf {Q}_{mathbf {1,2}}) is not Q-universal. This provides a positive answer to the following long standing open question: Is there a quasivariety that is relatively finite-to-finite universal but is not Q-universal?

We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational axiomatisation for the class of algebras representable by partial functions. As a corollary, the same equations axiomatise the algebras representable by injective partial functions. For complete representations, we show that a representation is meet complete if and only if it is join complete. Then we show that the class of completely representable algebras is precisely the class of atomic and representable algebras. As a corollary, the same properties axiomatise the class of algebras completely representable by injective partial functions. The universal-existential-universal axiomatisation this yields for these complete representation classes is the simplest possible, in the sense that no existential-universal-existential axiomatisation exists.